BGE Video Textures not showing up in exported .app
The VideoTexture module expects external files. It does not work with internal data blocks unless explicitly provided by Python call.

Therefore "Pack External Files" will not solve your issue. It increases the size of your distribution as you need to distribute the external files (as separate files) anyway.

I suggest following:


Perform "Make Paths Relative" before distributing
Ensure to distribute your VideoTexture source files where they are (relative to the start blend file).